What moral (not legal) obligations does the United States have in International Human Rights matters?

As well as we are able, we should seek to prove shelter, food, medical care, etc... for refugees and even those trapped in war zones. What we shouldn't do is get involved in the internal affairs of foreign nations. If we are certain that genocide is taking place we should work to build an international coalition that is capable of ending that genocide and protecting human life as quickly as possible.

We should also be willing to condemn governments that fail to respect human rights, or which do not have representative governments and impose sanctions equally on all repressive governments and their officials.

The exact same moral obligations every human being in every other country has... To be good, caring, compassionate human beings who care for others and work toward the betterment of our species. Anything less is inevitable societal collapse or species extinction..

Not all human rights issues can be solved by external forces. We also have to be careful about providing aid to governments that do not really care about their people. If they did care about them the crisis may not be occurring min the first place.

Since all human rights issues do not have the same cause and consequently the same resolution it is a hard question to answer directly. We have no obligation in these matters, we do them because we want to relieve suffering where possible.

Suffering is not always easy to relieve. In many cases, the human rights issues are created buy the very governments of the countries these people live in. Without taking out the government, you cannot really resolve the issue. Taking out government leaders and allowing the power vacuum to take effect does not usually end well. Take Libya for a recent example.